HIAS-E-24

Selective Incentives and Intra-Group Heterogeneity
in Collective Contents
Shmuel Nitzan
Department of Economics, Bar Ilan University, Ramat Gan, 52900 Israel
Hitotsubashi Institute for Advanced Study, Hitotsubashi University

Kaoru Ueda
Faculty of Economics, Nanzan University, Nagoya, Aichi 466-8673, Japan

March 31, 2016

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Selective Incentives and Intra-Group Heterogeneity
in Collective Contests 1
byShmuel Nitzan2 and Kaoru Ueda3

Abstract
A group taking part in a contest has to confront the collective-action problem amongits members and devices of selective incentives are possible means of resolution. We
argue that heterogeneous prize-valuations in a competing group normally prevent
effective use of such selective incentives. To substantiate this claim, we adopt cost
sharing as a means of incentivizing the individual group members. We confirm thathomogeneous prize valuations within a group result in a cost-sharing rule inducing thefirst-best individual contributions. As long as the cost-sharing rule is dependent only
on the members' contributions, however, such a first-best rule does not exist for agroup with intra-group heterogeneity. Our main result clarifies how unequal prize
valuations affect the cost-sharing rule and, in particular, the degree of cost sharing.
The results are related to the fact that heterogeneous valuations of the prize in a groupcause inappropriate realization of voluntary contributions, a situation known as the“exploitation of the great by the small.”JEL Classification: D70, D71, D72
Keywords: collective contest, selective incentives, intra-group heterogeneity, costsharing, elasticity of marginal costs, the “exploitation of the great by the small.”1

The first author is grateful to HIAS for enabling the completion of this study in a relaxed yet
stimulating environment.

2

Department of Economics, Bar Ilan University, Ramat Gan, 52900 Israel and Hitotsubashi Institute ofAdvanced Study, Hitotsubashi University. e-mail: nitzans@biu.ac.il
3

Faculty of Economics, Nanzan University, Nagoya, Aichi 466-8673 Japan. e-mail: k-ueda@nanzanu.ac.jp

11. Introduction
In a collective contest the contestants for a prize are groups. Applications of suchcontests include confrontations between labor unions and the employers, ethnic or
religious conflicts, military conflict between countries or allies of countries,
promotional competitions by firms with marketing activities, a championship by
sports teams, competition among academic institutes on quality-based recognition or
on financial support, and so on4. Our main objective is to study how intra-groupheterogeneity of a competing group relates to the choice of selective-incentive
devices.
When individuals win or lose the prize as a group, they work as a teamsharing a common aim. In such a situation, the individuals are usually tempted to be
free-riders while considering contribution to the teamwork to enhance the groupwinning probability. This tendency results in a typical collective-action problem, as
argued by Olson (1965, 1982)5. Collective contests could be viewed as a number of
intra-group collective-action problems embedded in a competitive environment.
Olson argues that the collective-action problem can be amended by “selective
incentives,” - incentives applied selectively to individuals depending on their actions6.
This conjecture is applicable to the competing groups in a contest, and we may
imagine the following thought experiment.
Suppose that there exists a first-best device of selective incentives which
fully resolves the collective-action problem; it permits the group to induce theindividual members to choose the collectively optimal contributions for the groupobjective. If the group can bring about commitment to this device, it will certainly doit. All of the paradoxical results associated with the collective-action problem would
then vanish. Although each group consists of many individuals, their contributions arecompletely coordinated for realizing the group objective. Such a collective contest
could not be distinguishable from a contest by individuals who use multiple inputs tocompete on the prize. This is of course an idealized story on the power of selective
4

For more examples of contests in general, see Konrad (2009). On the basic theory of contests, see
Hillman and Riley (1989) and Cornes and Hartley (2005).
5
As similar concepts to free-riding, we could count shirking and social loafing (Kidwell and Bennett.
1993). The former is a term used in the context of the economics of organization, and the latter ismainly used in studies of social psychology. All three concepts concern individuals withholding effort
in a group. Such overlapping of concepts in the different areas stresses the substantial role played by
the free riding problem in determining the performance of a group.
6
Olson’s conjectures are neatly arranged and evaluated by Sandler (1992). For two recent surveys on
the development of research on collective-action problems see Pecorino (2015) and Sandler (2015).

2

incentives; the widely observed collective action problems in the real world suggestthat they cannot be so effective. But we could use the above scenario as a benchmark
to examine how collective contests can be different from those played by individuals.
Studying the determinants of the ideal work of selective incentives, we get new
insights regarding the relevant theoretical factors, such as the characteristics of agroup that are important to secure advantage in a contest, the devices ensuring
effective selective incentives, and so on.
This paper emphasizes intra-group heterogeneity as a significant adverse
factor to the working of selective incentives. The individuals in a group are usuallysituated at different positions, politically, economically and sometimes ethnically or
culturally. Such heterogeneity could prevent the individual group members fromreaching a consensus on the value of the contested prize, so they naturally havedifferent prize valuations7. Olson (1982) conjectures that heterogeneity of a groupadversely affects the effectiveness of selective incentives, but his argument does notprovide any clue on the question how the form of selective incentives is bent undersuch less availability. We present an answer to this question focusing our attention on
the inefficient pattern of individual contributions caused by intra-group heterogeneity;from the collective viewpoint, an individual with a high valuation of the prize puts toomuch effort. The existing literature refers to the problem as the “exploitation of thegreat by the small,” the term originated by Olson (1965). We will show that selective
incentives can aggravate such exploitation.
For this purpose, we use a modified model of Nitzan and Ueda (2011). In
their model, each group consists of homogeneous individuals with the same valuationof the prize and the same form of effort cost functions. The selective incentives are
given by way of prize-sharing rules according to the relative effort of the groupmembers; part of the prize has the form of a private good and more is distributed outof this part to an individual contestant contributing more. They consider a collective
contest in which each competing group can commit (before the contest starts) to a
prize-sharing rule. They also assume that such a rule applied in a group is not
7

Intra-group heterogeneity is not an old topic in the literature of collective contests. Baik (2008)
examines a model of collective contests for group-specific public-good prize, assuming linear costfunctions, i.e., constant marginal costs. Esteban and Ray (2011) study a model of ethnic conflict withtwo parameters of intra-group heterogeneity: ethnic radicalism and income. Epstein and Mealem(2009) and Ryvkin (2011) examine how the composition of individuals in competing groups affects theequilibrium effort levels. Nitzan and Ueda (2013) clarify how the form of effort cost functions
determines the relation between intra-group heterogeneity and the equilibrium group effort.

3

verifiable for outsiders, because its application is only observed by the groupmembers. That is, the device of selective incentives in each group is endogenous8 andunobservable. In this setting, there exists for each group a first-best prize-sharing rulemaximizing the utilitarian group-welfare, which depends on the group size, thevaluation of the prize and the form of the effort cost function. It is possible to provethat the equilibrium prize-sharing rule of each group9 actually coincides with thisrule. Hence, their model illustrates the possible realization of the idealized story on
selective incentives.
We introduce two new important changes to the model. The first is theexistence of intra-group heterogeneity. Unfortunately, however, it is not easy toanalyze a model of prize-sharing with intra-group heterogeneity and, in particular, tocharacterize the equilibrium prize-sharing rules. It is especially so when the effort costof an individual is non-linear, and such non-linearity is essential to get generalinsights on the collective-contest problem.10 So we introduce a second variation;
instead of prize-sharing rules, our model assumes that the competing groups use costsharing rules as a means of selective incentives.
It has been recently pointed out by Vázquez (2014) that commitment to atransfer rule of the costs among individual group members can work as a substitutefor a prize-sharing rule. Actually, once we notice that cost-sharing makes theresources of the individuals in a group a common pool resource, it is not surprisingthat such a device enhances their activity levels. Transfer schemes within a groupdepending on the sacrifice that individuals make to enhance their group’s common
8

See Baik (1994), Lee (1995), and Ueda (2002) for early developments of the models withendogenous prize sharing. It should be noted that higher selective incentives are not necessarily better.
When the members are rewarded for their effort, each member’s effort has a negative externality forthe others because their shares are cut. The result might be an excessive group effort. See Sen (1966).
9
It is characterized in Theorem 1 (ii) of Nitzan and Ueda (2011).
10
Several impressive results derived from contest models with linear effort costs are not necessarily
preserved under non-linear cost functions. For example, Esteban and Ray (2001) reveal that collective
contests for a pure private-good prize with linear effort cost functions belong to a special case where
the “group size paradox” is always obtained, i.e., a smaller group attains a higher win probability. They
prove that the paradox is overturned if the elasticity of marginal effort costs is large. Another puzzling
possibility is that an individual with a higher prize valuation can get a lower expected payoff in a group(it could be called a strong version of “the “exploitation of the great by the small”). This is a normalcase in contests with linear effort costs, unless the largest value of the valuations is very prominent.
The reason is, as shown by Baik (2008), that in this case only individuals with the largest valuation of
the prize in a group put effort, and all the other group members become pure free riders (see also Lee(2012) for interesting related results). But Nitzan and Ueda (2013) point out that such exploitation is
impossible if the elasticity of marginal effort costs is large. We will present another example of thepeculiarity of the model with linear cost functions in Subsection 5.2. These examples imply that weshould be careful regarding the robustness of the results obtained under linear costs.

4

interest can work as selective incentives. An individual who contributes more canshift more cost to the others relative to the cost imposed on him/her by the others, andas a result, get a net transfer.
With the device of cost-sharing, it is possible to reproduce the same ideal
story on the effectiveness of selective incentives as in Nitzan and Ueda (2011); if theindividuals in a group are homogeneous, full sharing of the costs among the groupmembers is the first-best cost-sharing rule maximizing the utilitarian group-welfare
and the equilibrium rule chosen by each group coincides with this rule. This case cantherefore be used as a benchmark to check the effects of intra-group heterogeneity.
Furthermore, we can confirm that the model with cost-sharing has an easily tractableequilibrium, even when allowing intra-group heterogeneity and non-linear effort
costs.11 Assuming cost-sharing, instead of prize-sharing, as the incentive devices forcompeting groups, we have a handy and workable model for investigating therelationship between intra-group heterogeneity of prize valuations and theeffectiveness of selective incentives.
While full sharing of the costs is the first-best rule for a group of
homogeneous individuals, we also find that intra-group heterogeneity of the prize
valuations requires the first-best cost-sharing rule to impose different degrees of cost
sharing on the group members12. That is, the group needs to set a discriminatory(personally varied) cost-sharing rule to induce such contributions. This finding points
to a serious limitation intra-group heterogeneity imposes on a competing group. A
device of selective incentives normally operates uniformly and impersonally, i.e., it
does not discriminate individuals by their names. It specifies a reward that hinges only
on the individual’s behavior. This means that usually a group would not be able toimplement the first-best incentive device.
If a group has to set a uniform cost-sharing rule to individuals withheterogeneous valuations of the prize, how does intra-group heterogeneity affect theextent of cost-sharing or selective incentives? The answer to this question is our main
result. We identify a condition that determines to which direction unequal valuations
11

When the contested prize is a group-specific public good, the device of prize sharing cannot beapplied, at least in a straightforward way. It also does not work well when the prize is a group-specificcommons openly accessible to all members of the winning group, which is the case studied in Nitzan
and Ueda (2009). Even in such cases, a group could commit to a cost-sharing rule that imposes partial
sharing of the cost of the members’ sacrificed efforts.
12
The formal definition of the first-best cost-sharing rule is given in subsection 3.2.

5

of the prize within a group shift the degree of cost-sharing. If the relative rate ofchange of the marginal effort costs is decreasing, the degree of cost-sharing is reduced
by intra-group heterogeneity. If it is increasing, the cost is fully shared, but it cannot
induce the first-best contributions for the group. As argued above, the result is due tothe inefficient realization of the voluntary contributions induces by heterogeneous
valuations of the prize.
In the next section, our basic model is introduced. It treats the case where
each competing group applies a uniform cost-sharing rule to its members. Section 3
deals with first-best cost-sharing rules. We start with the analysis of a collective groupcontest where the individuals in a group act cooperatively for enhancing their group’s
interest (we will call it a contest by fully regulated groups). Such a contest is aconvenient tool to treat the cases in which the first-best selective incentives areavailable. Section 4 is the main part of the paper; it examines the relation between theform of equilibrium cost-sharing rules uniformly applied to all group members andintra-group heterogeneity in prize valuations. Section 5 focuses on the case of
constant elasticity of marginal effort costs, a convenient special case to make ourmain story transparent. The conclusions are presented in Section 6. All the proofs
appear in Section 7.
2. The Model
2.1 Group contests with cost-sharing
Consider m groups competing for a prize. The number of individuals belonging togroup i is denoted by Ni. Each person is assumed to be risk-neutral, who individuallyand simultaneously decides how much to contribute to enhance the win of his group.
The individual contributions are aggregated in every group, and the group probabilityof winning the prize is determined depending on those aggregated group efforts.
Assumption 1: The win probability of group i is given by Ai , where Ai is the effort ofA

group i, and A = ∑mj=1 A j is the total amount of effort by all competing groups. The
group effort of i, Ai, is given by Ai = ∑kN=1 aik ,13 where aik ≥ 0 denotes the efforti

13

See Kolmar and Rommeswinkel (2013) for contests with more general ways to aggregate efforts byindividuals in a group.

6made by member k of group i.
All members of a group have the same form of the effort cost function ci ,i.e. member k of group i has the cost ci (aik ).
Assumption 2: The cost function of individuals in group i, ci, is a thrice differentiable
function with ci (0) = 0 , ciʹ (a ) > 0 and ciʹʹ(a ) > 0 for all a > 0 . Also, lim a→0 ciʹ(a) = 0 .14
The valuation of the prize can be different among the group members, reflecting theirdifferent positions within the group. The stake for the kth individual belonging to theith group is denoted by vi k > 0 , which could be interpreted as the individual’svaluation of the prize15. Without loss of generality, we can set vi1 ≤ ! ≤ vi N . We willi

use the notation Vi = ∑kN=1 vi k . The distribution of the members’ stakes in the contesti

can be represented by the stake vector of group i, v i = (vi1 ,…, vi N ) .
i

Assumption 3: Every individual group member knows the stake vectors of all thegroups, but not the prize valuation of specific individual members (except himself) in
his or in any other group16.
Now, let us introduce and discuss the device of selective incentives in ourmodel. It takes the following form of uniform cost-sharing in each group. We assume
that part of the cost of the members’ contributions is shared within the group.
Formally, the following class of cost-sharing rules is available for each group.
Assumption 4: Group i can specify the value 0 ≤ δ i ≤ 1, the ratio of the effort cost of
every member compensated by making equal payback transfers that sum up to δ i of
14

The assumption lim a→0 ciʹ(a) = 0 excludes the possibility of non-contributors. As Nitzan and Ueda

(2013) argue, such possibilities have important implications on the relation between the groupperformance and intra-group heterogeneity in terms of the stakes. The main concern of this paper is,
however, the relation between the equilibrium cost-sharing rule and intra-group heterogeneity. Hence
ignoring the possibility would be justified by the transparency of the analysis.
15
We may interpret it as the valuation of a mixed private-public-good prize. Then vik is a function of
two variables, the kth individual’s share of the private good prize and the public good prize.
16
In other words, every individual does not know the ranking of other individuals in terms of thevaluation of the prize.

7

the total exerted efforts. In other words, the cost of individual k belonging to group i
has the form∑
(1 − δ )⋅ c (a ) + δ
i

i

ik

i

Ni

c (aip )p =1 iNi.

(1)

We may call a value of δ i the degree of cost-sharing. Since each individual is riskneutral, the utility of member k of group i is given byNi⎧
c (aip )⎫⎪∑Ai⎪
p =1 iEU ik = vik − ⎨ (1 − δ i )⋅ ci (aik ) + δ i
⎬.
A
Ni
⎪⎩
⎪⎭

(2)

Assumption 5: The degree of cost-sharing is determined in every group prior to thecontest to maximize the utilitarian group welfare (i.e. the sum of the expected utility(2) of all members of the group).
This decision on the cost-sharing in each group could be considered to be made (andimplemented after the contest) by a benevolent group leader.17
After observing the cost-sharing rule chosen by his own group, each member
chooses the effort level individually. The higher the degree of cost-sharing in a group,
the larger the transfer to an individual as the return of the contribution.18 When
δ i = 0 , no part of the cost of individuals in a group is shared, and there exists noselective incentives. If 0 < δ i < 1, the enforced cost-sharing is partial. When δ i = 1 ,the cost is fully shared. We assume that it is the highest degree of cost-sharing19.

17

Such an interpretation could be justified if the position itself is the intrinsic objective of the leaders,
and the nomination requires the consensus of the group members. Notice that the leaders must know
the distribution of the valuations of the prize in all groups. Another possible interpretation is that δ i is
accepted by the individuals belonging to a group, all of whom agreeing to the utilitarian value
judgment.
18(c (a ) − ci (aip )) , which implies that an
Notice that the function (1) can be written as c (a ) − δ ∑ p ≠ k i ik
i
ik
i
Ni
individual who makes relatively larger contributions than others in the same group gets a net transfer.
The amounts get larger as the value of δ i rises. This is the way how cost-sharing rules work as adevice creating selective incentives.
19
One may further consider the case of over cost-sharing, i.e. δ i > 1 . In such a case, the amountexceeding the real costs is redistributed in the group. We exclude this possibility by two reasons: first,
we are not aware to real examples of over cost-sharing. Second, it is ambiguous how far over costsharing can be advanced. The marginal cost of the individuals could be even less than or equal to zero8At this stage, we introduce the assumption of unobservable sharing rules.
Assumption 6: The decision on the degree of cost-sharing in a group is unobservableby those belonging to other groups.
This assumption is a variant of those made by Baik and Lee (2007) and Nitzan andUeda (2011), who consider the determination of prize-sharing rules by competinggroups. Both prize-sharing and cost-sharing result in redistribution within a group,and they are applied only with respect to insiders. Such inside rules are changeable bynotification only to the group members, and the changes could be made secretly, thatis, without informing other groups. Even if a group sharing rule is openly announced
to outsiders, they would hardly believe that the announced rule is the final one thegroup has really committed itself to. Hence, it is doubtful whether the redistribution
rules applied to the insiders can work as strategic devices for the opposing groups20.
Even for cases where such observable commitment is possible, checking what
happens if the sharing rules are unobservable is meaningful to isolate the purestrategic effects they have.
We do not apply, therefore, the usual two-stage-game formulation, where in
the first stage the cost-sharing rules are committed in each group and are publicly
known, and in the second stage the prize is contested. Instead, the contest under aconfiguration of cost-sharing rules in the competing groups is not a proper sub-gamein our model. Every member in a group needs to infer the sharing rules of the othergroups when he decides how much to contribute in order to enhance the winning of
his group. We therefore adopt the perfect Bayesian equilibrium notion as the solution
concept of our model. Its precise description is given in the next subsection.

2.2 Equilibrium
The solution concept we use is basically the same as that in Nitzan and Ueda (2011).
It is assumed that each player can use only pure strategies in equilibrium, and thepossibility of randomization in the information sets is omitted. To characterize perfectby setting δ ≥ N i , which makes the whole model intractable.
iNi − 1
20This problem is firstly pointed out by Katz (1991).

9

Bayesian equilibrium, we need to describe the beliefs in the information sets of theplayers. Since the decisions on the cost-sharing rules by the leaders are madesimultaneously at the beginning of the game, their beliefs are trivially given; they
believe that all leaders choose the equilibrium cost-sharing rules.
Hence, let us consider the beliefs and the strategies of individuals who makeeffort for the contest. Each information set of the kth individual belonging to group i
can be indexed by a value of δi corresponding to the cost-sharing rule announced by
the group leader. The individual cannot distinguish the nodes at which different
sharing rules are chosen in other groups. A strategy of the member is, therefore,
described as a function of only one variable δi, which is denoted by aik (δ i ) . Also,
this individual’s belief µik with respect to the other groups’ cost-sharing rules candepend on the value of δi. Then µik(δi) is a probability measure defined on [0, 1]m −1 ,the space of possible configurations of the cost-sharing rules in the other groups
δ −i = (δ1,!, δ i −1, δ i +1,!,δ m ) .

Pick a profile of the sharing rules δ1* ,!, δ m* and individual decisions on
effort,(a (δ ), µ (δ )) , for all δ ∈ [0, 1] , j = 1,!, m and h = 1,!, N . Let us
*
jh

j

*
jh

j

j

j

consider what conditions must hold if it is an equilibrium configuration. The expectedutility of individual k belonging to group i at the information set indexed by δ i is
calculated asNi⎧⎛⎞
c (aip (δ i ) )⎫⎪∑Ai* (δ i )⎪
p =1 i
*
⎜
⎟
(
)
(
)
(
)
vik ⋅ ∫
µ
(
δ
)
d
δ
−
1
−
δ
⋅
c
a
(
δ
)
+
δ
⋅
⎨
⎬,
−i
i
i
ik
i
i
⎜ ∑ A*j (δ j ) + Ai* (δ i ) ⎟ ik i
N
i
⎪⎩
⎪⎭
⎝ j ≠i⎠where A*j (δ j ) = ∑ Np=1 a*jp (δ j ) . At the information set indexed by δi*, which lies on thejequilibrium path, µik* (δ i* )({δ −i = δ −*i }) = 1 holds by the requirement of consistency; theindividuals must correctly infer the sharing rules in the other groups, given thestrategies of the leaders.
If we apply the “no-signaling-what-you-don’t-know” condition of Fudenberg
and Tirole (1991), we can also restrict the beliefs of the individuals in the information
sets outside of the equilibrium path. This condition requires that choices of a groupleader should not inform the group members anything about what she doesn’t know.21
Since individuals in each group know that their own group leader cannot see any
21

The formal condition is given in Definition 6.1 of their paper.

10

change of cost-sharing in the other groups, they should infer that her deviations tellnothing new about this matter. At an un-reached information set, therefore, theassociated individual should keep the same belief as that held if the equilibrium groupsharing rule is announced. The equilibrium belief by the kth member of group i, µik*,
must satisfy the conditionµik* (δ i )({δ −i = δ −*i }) = 1, for all δ i ≠ δ i* .

We have described our solution concept for the model, a pure-strategy
perfect

Bayesianequilibrium

(with

the

“no-signaling-what-you-don’t-know”

condition). The following Lemmas are useful for characterizing equilibrium.
Lemma 1. The equilibrium contribution by individual k of group i who is aware of
the cost-sharing rule δ i (i.e. at the information set indexed by δ i ) is described by astrictly increasing differentiable function aik* (δ i ) defined by the following equation:∑

A*j (δ *j )⎧
δ ⎫vik − ⎨(1 − δ i ) + i ⎬ ⋅ ciʹ aik* (δ i ) = 0, k = 1,!, N i ,Ni ⎭⎩∑ j ≠i A (δ ) + A (δ i )

{

j ≠i

*
j

*
j

(

}

2

*
i

(3)

)

given the other groups’ equilibrium cost-sharing rules δ *j ,j ≠ i.

Lemma 2. (a) For each level of efforts by the other groups ∑ Aj , group i canj ≠iattain the aggregate group effort Ai if and only if it belongs to the closed interval[A (∑ A ), A (∑ A )], where A (∑ A ) and A (∑ A ) are uniquely given
L
i

j ≠i

H
i

j

j ≠i

j

L
i

j ≠i

∑

j ≠i

H
i

j

j ≠i

j

by the equations⎡Ni∑k =1 (ciʹ )−1 ⎢(⎤⎥ = AiL ∑ A j , and
v
2 ik
j ≠i
⎥
⎦

Aj

(

⎢ ∑ A j + AiL ∑ A j
j ≠i
j ≠i
⎣

(

))

⎡
∑ j ≠i A j∑k =1 (ciʹ ) ⎢⎢ N i ⋅∑ j ≠i Aj + AiH ∑ j ≠i Aj
⎣
Ni

−1

(

(

)

⎤
v ⎥ = AiH ∑ j ≠i A j .
2 ik
⎥
⎦

(

))

)

(b) For each level of group effort Ai > 0 mentioned in (a), the allotment of thecontribution in the group is given by the functions aik (Ai ; vi ) ’s defined by thefollowing equations:

1

Ni

1

∑ a (A ; v ) = A and v ⋅ cʹ(a (A ; v )) = v ⋅ cʹ(a (A ; v )) for all k = 1,!, Ni .
k =1 ik

i

i

i

i

ik

i

i

ik

i

i1

11

i1

i

i

(4)
Lemma 1 shows that there exists an one-to-one relation between the degreeof cost-sharing and the equilibrium contributions by individuals in a group, given theefforts of the other groups, A*j (δ *j ), j ≠ i . Then the leader of a competing group cancontrol the level of group effort via the choice of the degree of cost-sharing.22 Lemma
2 (a) specifies the range of group effort that the leader can attain, and 2 (b) specifies
how such a level of group effort is bore within a group.23 We can therefore define thefunction specifying how much cost is sacrificed in the group if the leader wants to
induce a given level of group effort;N

Ei ( Ai ; v i ) = ∑k =i 1 ci (aik ( Ai ; v i )) .
It does not necessarily coincide with the minimum sum of the group members’ effort
cost to induce a given aggregate effort Ai, because they voluntarily and individuallychoose their contributions. So we refer to Ei (Ai ) as the distorted group cost functionof i. With this function, the utilitarian group welfare of group i can be represented as afunction of group efforts A j , j = 1,!, m , as follows;

Ai

∑

m

A
j =1 jVi − Ei ( Ai ; v i ) .

(5)

The next lemma on the distorted group cost function is useful.

vikciʹʹ(aik ( Ai ; v i ))∂N
Lemma 3.
Ei ( Ai ; v i ) = ∑k =i 1 ciʹ (aik ( Ai ; v i )) ⋅v
N∂Ai
∑ p =i 1 cʹʹ(a (ipA ; v ))i
ipii
holds for Ai > 0 . Furthermore, lim Ai →0(6)∂
Ei ( Ai ; v i ) = 0 .
∂AiNow, consider a configuration of group efforts A*j , j = 1,!, m such that Ai*
is a solution of the maximization problem:
22

Notice that the individuals in a group retain the same belief on the degrees of cost-sharing (and thegroup efforts ) in the other groups, when being told that different cost-sharing rules are applied in theirown group.
23
The representations of the contribution by an individual in the two lemmas can be related by theequation: aik Ai* (δ i ); v i = aik* (δ i ) .

(

)

12

max Ai ≥0

AiVi − Ei ( Ai ; v i )∑ j ≠i A*j + Ai

subject

to

(

)

(

)

AiL ∑ j ≠i A*j ≤ Ai ≤ AiH ∑ j ≠i A*j ,(7)

for all i = 1,!, m . Then we can find a profile of the sharing rules δ 1* ,!, δ m* describedby the equations ∑kN=1 aik* (δ i* ) = Ai* , i = 1,!, m , using the function in Lemma 1. The
iabove constraint on group effort corresponds to the constraint on the degree of costsharing, 0 ≤ δ i ≤ 1. Hence it is a profile of cost-sharing rules in the pure-strategy
perfect Bayesian equilibrium. EachAi* is the group effort in equilibrium.

Conversely, if A*j , j = 1,!, m are group efforts in equilibrium, then they are thesolution of the above maximization problems. In the rest of the paper, we will often
resort to this characterization of equilibrium group efforts.
To simplify the analysis, let us introduce the following condition.
Regularity Condition: The distorted group cost Ei ( Ai ; v i ) is convex in Ai.
It is easy to confirm that the condition is always satisfied in the case of intra-grouphomogeneity, i.e. all individuals in the same group have the same valuation of the
prize; vik = vi1 for all k = 1,!, Ni . When a group exhibits intra-group heterogeneity,
one sufficient condition for convexity is that the relative rate of change of theʹmarginal effort cost ciʹʹ(a ) and its logarithmic derivative ⎛⎜ log ciʹʹ(a ) ⎞⎟ are decreasing⎜⎝ciʹ (a )ciʹ (a ) ⎟⎠in a. Under these conditions, individuals with higher valuations of the prize in a group
have positive∂ 2 aik ( Ai ; v i )∂Ai

2and those with lower valuations have negative∂ 2 aik ( Ai ; v i ) . It is sufficient for our regularity condition to hold. The convenient∂Ai

2special case of constant elasticity of marginal effort costs, in which ci (a ) =Ki 1+α ia
1 + αifor all i = 1,!, m , where α i > 0 and Ki > 0 are positive constants, satisfies these
sufficient conditions. Applying the regularity condition, the existence and uniquenessresults are derived.

13

Proposition 1. If the above regularity condition holds, then there exists a uniquepure- strategy perfect Bayesian equilibrium in our model of group contest with costsharing.
3. The Possibility of First-Best Cost-Sharing
3.1. Contests by fully regulated groups
If individuals in a group perfectly obey the group leader, she could directly assign
them the contributions that maximize the objective function of the group. In
equilibrium of such a contest by fully regulated groups, each group leader sets theefforts of his group members such that
N
Ni
⎛
⎞
a
∑ j =i 1 aij∑N⎜⎟
j =1 ij
vik − ci (aik )⎟ =Vi − ∑k =i 1 ci (aik )∑k =1 ⎜⎜Ni
Ni
⎟ ∑ A j + ∑ aij
A +
a
j ≠i
j =1
⎝ ∑ j ≠ i j ∑ j =1 ij⎠
Ni

is maximized with respect to aik ≥ 0, k = 1,!, N i , given the other groups’ efforts. It iseasy to see that the solution must minimize the aggregate group cost ∑kN=1 ci (aik )igiven the equilibrium aggregate group effort. So we can apply the popular two-stage
approach to a maximization problem: define the group cost function

{

N

}

N

Ei* ( Ai ) = min ∑k =i 1 ci (aik ) ∑k =i 1 aik = Ai , aik ≥ 0, k = 1,!, N i .
The first-order condition for the solution of this minimization problem isciʹ(aik ) = ciʹ(ai1 ) for all k = 1,!, N .

(8)That is, the equalization of the individuals’ marginal effort costs. Since the marginal
effort cost function is strictly increasing, it requires equal contributions by all
individuals in the group.
Hence Ei* ( Ai ) = N i ⋅ ci ⎛⎜ Ai ⎞⎟ and the equilibrium choice by the group leader of
⎜N ⎟
⎝ i⎠

a fully regulated group is simplified to choosing the group effort Ai ≥ 0 to maximize⎛A ⎞Ai
Vi − N i ⋅ ci ⎜⎜ i ⎟⎟ ,∑ j ≠i Aj + Ai
⎝ Ni ⎠
given the other groups’ efforts. The first-order (necessary and sufficient) condition forthe solution of this problem is

∑

j ≠i

Aj

(∑ A + A )

2

j ≠i

j

i

⎛A ⎞
Vi − ciʹ⎜⎜ i ⎟⎟ = 0 .
⎝ Ni ⎠

14(9)Viewing each group leader just as an individual competing on the prize, we can treat
the contest by fully regulated groups as a contest by individuals with convex
technologies, which is considered by Cornes and Hartley (2005) and the existence of
unique equilibrium on group efforts Ai’s is guaranteed. Each individual in a groupcontributes 1/ Ni of the equilibrium group effort.
3.2. Efficiency of cost-sharing rules
Formally, the first-best cost-sharing rule induces the same equilibrium contributionsof individuals as those in a contest by fully regulated groups. That is, a cost-sharing
rule that induces the contributions satisfying conditions (8) and (9) in equilibrium isfirst-best. If there exists such a cost-sharing rule for each group, every group leader
will choose it thus ensuring that the equilibrium group efforts coincide with those of
the unique equilibrium of the contest by fully regulated groups. Such a situation is
obtained in a group consisting of homogeneous individuals.
When the valuation of the prize is common among the individuals belonging
to group i, we can write vik = Vi for all k = 1, !, N . Condition (4) then implies thatNi

the marginal costs of the individuals are equalized and they contribute equally 1/ Ni of
the group effort Ai. Now, let us set δ i = 1 and observe that condition (3) under suchfull cost-sharing always coincides with (9), because aik = Ai for all k = 1, !, N .
NiHence we have the following result (formally, it will be proved as a special case of
Proposition 4):
Proposition 2. When all individuals in a group are identical in their prize valuations,the cost is fully shared in equilibrium. This cost-sharing rule is first-best.24
When a group consists of individuals with heterogeneous prize valuations, any

24

In the model of collective contests with prize-sharing considered by Nitzan and Ueda (2011),
comparison between two homogeneous groups of the same size reveals that the group attaining thelower winning probability is the one dividing a larger part of the prize according to the relative effort
rule (See Nitzan and Ueda (2011) for the details). Interestingly, this seemingly strange pattern isequivalent to full cost-sharing in contests by homogeneous groups with cost-sharing. Once we noticethat the pattern is caused by the incentives depending not on the costs but on the prize, it becomes more
understandable; strong incentives are needed if the win probability is low because the reward for thecontribution is less probable.

15

uniform cost-sharing rule cannot be first-best. This result is directly obtained from thecomparison between conditions (4) and (8). The former condition requires that theratio of the marginal effort costs of individual group members is equal to that of theirprize valuations, as long as they contribute voluntarily. Hence the marginal costs arenever equalized, and condition (8) is not satisfied. Basically, this is how intra-groupheterogeneity of the prize valuations prevents the cost-sharing rules from resolving
the collective action problem.
We also see that it is an intrinsic problem for a group with intra-groupheterogeneity in prize valuations, because the marginal costs are not equal already inthe case where δ i = 0 , the case without cost-sharing. Misallocation of burdens in agroup is usually regarded as a fairness problem in the collective action literature, i.e.
the “exploitation of the great by the small” emphasized by Olson (1965) and Olson
and Zeckhauser (1966). But it also causes an efficiency problem as has been justshown, which is well-known in elementary microeconomics. And this problem is a
critical factor in determining the degree of cost-sharing.
The problem can also be viewed and analyzed as that of discriminating costsharing rules. Suppose that it is possible for group i to differentiate the size of theshared part of the cost among its members as follows. For individual k, the effort cost
is compensated at the individual-specific rate of δ ik ≥ 0 , k = 1,!, N i . The amount

∑

Ni
p =1

δ ip ci (aip ) needed for the compensation is collected equally from all the individual

group members. Then the cost of individual k has the form∑
(1 − δ )⋅ c (a ) +
ik

i

ik

Ni
p =1

δ ip ci (aip )Ni.

It could be called the sophisticated cost-sharing rule. With this rule, we obtain thenext result.
Proposition 3. By adequate choice of the sophisticated cost-sharing rule, group i can
induce the first-best contribution by its members, in the sense that their contributions
maximize the utilitarian group welfare, given the levels of aggregate group efforts of
the other competing groups. This first-best rule is characterized by the degrees ofcost-sharing16

v
1 − ikVi, k = 1,!, N i .
δ ik =
1
1−
Ni

(10)

When all individuals share the same valuation of the prize, δ ik = 1 holds for all
k = 1,!, N i . Otherwise, the degree of cost-sharing must be different from person toperson to induce the first-best contributions. Equation (10) implies that, undersophisticated cost-sharing, an individual with a lower prize valuation must receive alarger compensation. But such a requirement seems too stringent for a device of
selective incentives because usually it has to treat individuals uniformly and is notallowed to apply discrimination based on their names. A reward to an individual isprovided only according to his/her behavior. Why is it difficult to let incentive
schemes depend on specific individual characteristics? At least two plausible reasonscan be mentioned. First, differences in behavior among individuals are easily
observable, but this is not necessarily the case for individual characteristics such as
prize valuations. Second, different treatment of individuals depending on their names,
status, or other observable characteristics often involves political problems, eventhough such variable treatment is systematically related to different valuations. Hence
we assume that the available cost-sharing for a group is uniform and therefore theexistence of intra-group heterogeneity precludes the application of effective selective
incentives. The next section examines how this problem affects the choice of costsharing rules by a competing group.
4. Cost-sharing Rules under Intra-group Heterogeneity
4.1 Basic observations
To characterize the equilibrium rules of cost-sharing chosen by competing groups, it
is convenient to introduce a new variable, γ i = 1 − δ i + δ i . Using the first orderNi

condition (3) to determine each group member’s contribution, notice that it can beinterpreted as the discount factor of marginal effort costs reflecting cost-sharing.
Since this factor is strictly decreasing in δ i , we can argue on the equilibrium costsharing scheme of group i applying γ i instead of δ i . It takes the value 1 when none17

of the costs are shared ( δ i = 0 ), and the value 1 when the whole costs are sharedNi( δ i = 1). By using the properties of the distorted group cost function Ei (Ai ) and thefirst-order conditions for the maximization problem (7), the basic result on theequilibrium cost-sharing rules is derived;
Proposition 4. In equilibrium, the cost-sharing scheme chosen by group i satisfies theinequalitiesv
γ i ≤ (≥ ) ∑k =1 ik ⋅
Vi
Ni

vik

)
=
⎛v
⎞ ∑ cʹ(a )∑ ⎜⎜ cʹʹ(a )⎟⎟
⎝
⎠
Ni

p =1

if γ i > 1

Ni

∂Ei *
Ai ; v i
∂Ai

(

( )

ciʹʹ aik*
ip

i

Ni

p =1 i

*
ip

*
ip

(11)

( γ i < 1 ), where aik* is the equilibrium contribution by individual kbelonging to group i, and Ai* is the aggregate equilibrium effort of the group.25
Notice that the middle term of (11) is strictly less than one. Also notice that γ i < 1must hold if γ i is larger than this term. Thus γ i < 1 always holds and a direct
corollary of Proposition 4 is:Corollary 1. Under the pure strategy Bayesian perfect equilibrium of the model, atleast some degree of cost-sharing is implemented in every group.
Hence a competing group always adopts cost-sharing when such a device gives
selective incentives to its members. Also, Proposition 4 permits us to directly derive
the cost-sharing rule in case of intra-group homogeneity. Condition (4) requires that
contestants with the same stake will choose the same level of effort. This fact
combined with Proposition 4 proves that γ i = 1 , i.e. the costs of the individualNi

group members are fully shared.

25

The equation δ = 1 − γ ii

1− 1may be useful for interpreting (11).

Ni

18

4.2 Distortions of the degree of cost-sharing
Let us see what happens to the equilibrium cost-sharing rules under intra-group prizevaluation heterogeneity. To provide an answer, let ai be the effort level determinedby the equation ai = (ciʹ )−1 (Vi ) . Since an individual member of group i does not expect
to get more than his stake, in equilibrium every group member contributes less thanai . The following important result holds:

Proposition 5. Suppose that the valuations of the prize by individuals in group i are
not even.
(a) If the relative rate of change of the marginal effort costs, ciʹʹ(a ) , is strictlyciʹ (a )decreasing in a on the interval ( 0, ai ) , then in equilibrium 0 < δ i < 1 (i.e.,
1< γ i < 1 ).
Ni(b) If ciʹʹ(a ) is increasing in a on the interval ( 0, ai ) , then in equilibrium δ i = 1 (i.e.,ciʹ (a )

γi =

1 ).
Ni

The proposition reveals that intra-group heterogeneity works differently on the formof cost-sharing depending on a property of the relative rate of change of the marginal
costs. If it is strictly decreasing, partial sharing of costs is chosen by the group. If it isincreasing, the costs are still fully shared in the group.
The former case seems more probable because of the following reasons.
First, investigating conditions (4) from which the distorted group cost function isderived, we can see that ∂ a ( A ; v ) =
ik
i
i
∂Ai

ciʹ(aik ( Ai ; v i ))ciʹʹ(aik ( Ai ; v i )) . Also notice that the larger theN cʹ(a ( A ; v ))∑ p =i 1 cʹiʹ(aip (Ai ; vi ))
i
ip
i
i

valuation of the prize of an individual, the larger his contribution. If ciʹʹ(a ) isciʹ (a )increasing, an individual with a lower valuation of the prize would increase his
contribution more than others as the aggregate group effort rises. This seems rather
unusual. Second, a strictly decreasing ciʹʹ(a ) permits the elasticity of marginal effortciʹ (a )19

costs ciʹʹ(a ) ⋅ a to be constant, or to show up-and-down changes. This possibilityciʹ (a )seems more natural than the situation where the elasticity steadily rises, which isnecessary if ciʹʹ(a ) is increasing. We could therefore normally expect that intra-groupciʹ (a )heterogeneity in the valuations of the prize results in partial cost-sharing. Also notice
that, even if the relative rate of change of the marginal costs is increasing, the fullcost-sharing in this case cannot be the first-best. The contributions by the members
are still not equalized.
Why does heterogeneity of stakes within a group affect the choice of thedegree of selective incentives in the form of cost-sharing? Distributional concerns forthe net benefit of group members would play a minor role in the decisions of a groupleader because her objective is assumed to be the utilitarian group welfare. As wehave already argued in Section 3, cost-sharing rules uniformly applied toheterogeneous individuals fail to induce efficient contributions minimizing the sum ofeffort costs. Equal contributions by every individual in the group should be made, but
intra-group heterogeneity of valuations of the prize does differentiate the voluntary
contributions. The existence of such inefficiency and the direction of change
associated with a rise in a group effort determine how intra-group heterogeneity
affects the degree of cost-sharing.
If ciʹʹ(a ) is strictly decreasing, the inefficiency gets worse as the aggregateciʹ (a )effort grows because an individual with a higher valuation of the prize would more
rapidly enhance his contribution.26 This reallocation of contributions piles up the
extra burden of enhancing group effort, causing the leader to hesitate when
considering the application of a high degree of cost-sharing. In contrast, theinefficiency would be alleviated if ciʹʹ(a ) is increasing, because the lower the prizeciʹ (a )valuation, the larger the expansion of effort by the individual. The reallocation effectis now desirable to equalize the voluntary contributions, and the leader would like tociʹ(aik ( Ai ; v i ))ciʹʹ(aik ( Ai ; v i )) . If the proportional rate of marginalN cʹ(a ( A ; v ))∑ p =i 1 cʹiʹ(aip (Ai ; vi ))i
ip
ii
effort costs is strictly decreasing, this derivative is larger for an individual with a larger stake. If it isincreasing, the reverse relation holds.
26

As we have already noted,∂aik ( Ai ; v i ) =∂Ai20promote cost-sharing. The result is the highest possible degree of cost sharing.
This intuition is supported by confirming that the reallocation ofcontributions actually affects the level of the marginal group costs. Notice that1Nicʹ (aik ( Ai ; v i )) is the increment in the group’s costs provided that the individual∑
k =1 iNi

group members equally expand their contributions. Remembering that δ i < 1 isequivalent to γ i > 1 , we can combine the equality of the middle and the right terms
Ni

∂E i

(

)

p =1

( )

Ai* ; v i∂
A1
in (11) with Proposition 5 to get the following observations:< γ i = Ni
Ni
∑ i ciʹ a ip*

if

∂Ei *
(Ai ; v i ) cʹʹ(a )ʹʹci (a ) is strictly decreasing in a. 1
∂A
if i
is increasing in a.
= γ i ≥ Ni i*ʹ()
ciʹ (a )
c
a
Ni
ʹ
(
)
c
a
i
∑ i ipp =1

These observations are summarized in the next corollary.
Corollary 2. Suppose that the valuations of the prize by individuals in group i are nothomogeneous. If ciʹʹ(a ) is strictly decreasing in a,ciʹ (a )If ciʹʹ(a ) is increasing in a,ciʹ (a )∂1
Ni
Ei ( Ai ; v i ) >cʹ(aik ( Ai ; v i )).
∑k =1 i
∂Ai
Ni

∂
1
N
Ei ( Ai ; v i ) ≤∑ i ciʹ(aik (Ai ; v i )).
∂Ai
N i k =1

4.3 Intra-group heterogeneity and win probability
Our analysis proves that, when competing groups use cost-sharing rules as a device of
selective incentives, intra-group heterogeneity prevents effective use of them. Unlessdiscriminating and individualistic rules are available, a cost-sharing scheme induces
inefficient responses of the heterogeneous group members and the leader cannot
provide them with the first-best selective incentives. This problem would put such agroup in an inferior position to a homogeneous group. Under some conditions, theinferiority takes the plain form of lower win probabilities.
To study this issue, set Vi , the sum of the valuations of the individuals ingroup i, at a constant value. Let us consider a contest in which all individuals of groupi have the same valuation of the prize, i.e. every member has the valuation Vi of theNi

21

prize. Starting from the equilibrium of such a contest, we examine the effect ofchanging the stake vector of group i by making it unequal (but keeping Vi
unchanged). Does the equilibrium win probability of group i decline relative to theinitial equilibrium?
Even though inefficient contributions by individuals raise the group costs, theanswer to this question is not clear. Intra-group heterogeneity in valuations of the
prize causes an inefficient assignment of group efforts, but it does not necessarily
lower the equilibrium level of the group’s effort. As Nitzan and Ueda (2013) argue,
the intra-group heterogeneity itself (not the effect through inadequate selective
incentives) can enhance the group effort. Furthermore, strategic interaction among thecompeting groups can eventually enhance the group efforts. We can identify,
however, one of the cases in which intra-group heterogeneity has a negative effect on
the group’s prospect of winning.
Proposition 6. Let the average prize valuation in group i be constant.
(a) Consider the case where the relative rate of change of the marginal effort costs,ciʹʹ(a ) , is strictly decreasing in a. If the marginal effort costs of group i, ʹ , isci (a )ciʹ (a )convex, the equilibrium win probability of group i is reduced when its members
have heterogeneous prize valuations rather than the same prize valuation.
(b) Consider the case where ciʹʹ(a ) is increasing in a. If the marginal effort costs ofciʹ (a )group i is strictly convex, the equilibrium win probability of group i is reduced
when its members have heterogeneous prize valuations rather than the same prize
valuation.
It is widely argued in the literature of collective contests that, if the function of
marginal effort costs is strictly convex, then a competing group attains the maximum
win probability only when its members share the same stakes.27 But Proposition 6 is
not a simple repetition of the existing results because we are considering collective
contests with selective incentives. The prospect of a competing group is restricted byintra-group heterogeneity via selective incentives, particularly in the case where the27

See Section 5 of Nitzan and Ueda (2013).

22

relative rate of change of the marginal effort costs is strictly decreasing. Proposition 6
illustrates how the merit of selective incentives in collective contests is influenced byintra-group heterogeneity.
4.4 Intra-group heterogeneity with respect to cost conditions
In subsection 4.2, we have argued that inefficient contribution caused by intra-groupheterogeneity is an essential factor for inducing a group to deviate from full costsharing. To confirm this point, let us resort to the following argument. Keeping allother components of the basic model, individual valuations of the prize and the costfunctions are now changed as follows; the cost function of individual k who belongs
to group i becomes 1 ⋅ ci (aik ), and the valuation of the prize is just 1. Then, hisvikequilibrium choice of contribution satisfies the modified first-order condition
*
j

*
j

∑ A (δ )⎧δ ⎫ 1− ⎨(1 − δ i ) + i ⎬ ⋅ ⋅ ciʹ aik* (δ i ) = 0 ,Ni ⎭ vik∑ j ≠i A*j (δ *j ) + Ai* (δ i ) ⎩

{

j ≠i

(

}

2

)

(12)

which replaces the first-order condition (3) in the original model. It is obvious thatboth conditions (12) and (3) result in the same reaction to any given degree of costsharing, under any combination of the values vik’s. If intra-group heterogeneity can be
represented as a distribution of those vik’s, the relation between intra-groupheterogeneity and the induced contributions by cost-sharing is the same as in theoriginal setting. However, individual contributions no longer cause inefficiency here,
even though the vik’s are different within the group; heterogeneous individuals in agroup contribute differently because of their different technologies, i.e., costfunctions, but their marginal costs in equilibrium are now always equal. This modified
model can be referred to as the associated contest with intra-group heterogeneous
cost functions. In such an associated contest, how do the group leaders choose thedegree of cost-sharing?
The summed group cost in the associated contest is defined by

1
NEˆ i ( Ai ; v i ) = ∑k =i 1 ⋅ ci (aik ( Ai ; v i )),vikwhich equalizes the marginal effort costs of the individuals belonging to the group.
Since this is the function of the minimized group costs, we can see that the first-best
choice of aggregate group effort for the leader maximizes

23

Ai
N i − Eˆ i ( Ai ; v i ).
A
The following result is therefore obtained:
Proposition 7. Assume that the group cost function Eˆ i ( Ai ; v i ) is convex28. Then, theindividuals in a group fully share their costs in equilibrium of the associated contest
with intra-group heterogeneous cost functions.
That is, in the modified setting the cost is fully shared. It should also be clear that it is
the first-best cost sharing rule. How much contributions can be induced fromheterogeneous individuals is, therefore, not critical for a group leader in avoiding fullcost-sharing. The created inefficiency makes the difference.
5. Cases of Constant Elasticity of Marginal Effort Costs
If the class of effort cost functions is confined to the form of constant elasticity of
marginal costs, i.e. ci (a ) = K i ⋅ a

1+α i

1+ αi

( α i > 0 ), the model becomes surprisinglytractable. In this special case, the equilibrium discount factor of marginal effort costs,
γ i , is explicitly expressed with the individual valuations of the prize. Intra-groupheterogeneity can then be directly related to the degree of cost-sharing. Notice that, in
this case, our Regularity Condition actually holds. In addition, ciʹʹ(a ) = α iciʹ (a )aisdecreasing, which means that partial cost-sharing is expected. The marginal effort
cost function ciʹ (a ) = K i a α is convex if α i ≥ 1, and strictly convex if α i > 1.
i

5.1 Elasticity of marginal effort costs and the degree of cost-sharing
Some calculations using the conditions in (4) show that under constant elasticity of

28

The convexity holds in the cases of intra-group homogeneity and of constant elasticity of marginal
effort costs.

24

1vikα imarginal costs, aik ( Ai ; v i ) =

1

∑

Ni

Ai .

29

Also, the next lemma can be derived as av αi

p =1 ip

special case of Proposition 4. The equilibrium cost-sharing is partial in this case, as
Proposition 5 predicts.
Lemma 4. Assume that the effort cost function of every member of group i has theform of constant elasticity of marginal costs, ci (a ) = K i ⋅ a

1+α i

1+ αi

( α i > 0 , K i > 0 ). Then,the equilibrium cost-sharing rule has the following explicit form;
1

γi =

1
Ni

∑ p=1 vip

∑

Ni

αi

vik

+1
1

k =1

∑

Ni

.

(13)

v αi

p =1 ip

For an n-dimensional vector v = (v1 ,…, vn ) , the Lehmer mean with index q is definedby Ln (v, q ) = ∑n

vkq

k =1

∑

n
p =1

v

q −1
p

(Lehmer (1971)). By equation (13), the equilibrium cost-

sharing scheme in this special case can be represented as follows;⎛⎞1
LN i ⎜⎜ v i , + 1⎟⎟
⎝ αi
⎠.
γi =
Ni
∑ p=1 vip

(13′)

From the well-known properties of the Lehmer mean, we directly get the following
results on the relation between the cost-sharing schemes and the elasticity of marginal
costs.
Proposition 8. Assume that the effort cost function of an individual in group i has theform

ci (a ) = K i ⋅a1+α i
1+ αi( α i > 0 ). Then, given vi, the degree of cost-sharing inequilibrium is strictly increasing in α i . Furthermore, we have the two limit cases on
the equilibrium discount factor of marginal effort costs, lim α →∞ γ i = 1
i

Ni

and

1

29

αi
N
Notice that since ciʹ (a ) = K i ⋅ a , the conditions in (4) imply that aik = ⎛⎜ vik ⎞⎟ and ∑ i aik = Ai .
k =1⎜⎟
ai1 ⎝ vi1 ⎠αi

25

limα i →0 γ i ={max vi1 ,…, vi N i
Vi

}.30

Hence, given a distribution of stakes in a group, the higher the elasticity of marginal
costs, the nearer to full sharing the implemented cost-sharing. The value of limαi →∞ γ ishows that such propensity eventually results in the full sharing of effort cost. This
result is understandable in light of the argument regarding the reallocation ofcontributions presented in Subsection 4.2. When the elasticity of marginal effort costs
1

is constant, we have ∂aik ( Ai ; v i ) =∂Aivikα i
1∑

Ni. If the elasticity is high, the increments ofv αi

p =1 ip

contributions caused by enhancing selective incentives are almost the same for every
group member. In such a case, reallocation of contributions caused by expansion ofgroup effort is negligible, and the caused deviation from full cost sharing is small.
5.2 The peculiarity of linear cost functions
Our basic model assumes convex effort cost functions, and does not include the caseof linear effort cost function or constant marginal effort costs. But it is a limit case of
constant elasticity of marginal effort costs, because K i ⋅ a

1+α i

1+ αi

when α i = 0 . Actually, the value limα →0 γ i =i{max vi1 ,…, vi N iVi

has the form K i ⋅ a} in Proposition 8

coincides with the discount factor of the equilibrium cost-sharing rule in the case of
constant marginal costs. In that case, only the contestants with the largest stake in thegroup are active, as established in Baik (2008). Then the aggregate effort of group i,
Ai, is determined by the equation

A − Ai⋅ max vi1 ,…, vi Ni = γ i ⋅ K i .
A2

{

}

We can also show that the group effort maximizing the utilitarian group welfare isdetermined by the equation

A − Ai⋅Vi = K i .
A230

Since the right-hand-side of this equality is larger than or equal to the arithmetic mean of the stakesof the individuals, limα → 0 γ i is in fact larger than or equal to 1 .
i
Ni

26

Combining the above two equations, we can see that the first-best cost-sharing rule isinduced if the value of the discount factor of marginal effort costs is set at limαi →0 γ i .
The group leader will of course choose it.
A remarkable point in this argument is that the equilibrium cost-sharing rule
induces the effort maximizing the group welfare, even with heterogeneous valuationsof the prize. By virtue of constant marginal costs, the allocation of contributions
among individuals in a group does not matter. The inefficiency in contributions is notcaused, which in the general cases prevents the maximization of the utilitarian groupwelfare. With linear costs, therefore, the requirement that the sharing rule should be
dependent only on contributions causes no problem, and the sophisticated costsharing discussed in subsection 3.2 is redundant.31 This is another example of thepeculiarity associated with linear costs in the theory of collective contests.
5.3 Inequality of stakes within a group and cost-sharing
In the case of constant elasticity of marginal costs, we can find a much more regular
relation between a distribution of the stakes and the degree of cost-sharing than in thegeneral model. This is especially clear for the case of a quadratic cost function, i.e.
α i = 1, where γ i coincides with the Herfindahl-Hirschman Index (HHI);
LN i (v i , 2)

∑

Ni

v

p =1 ip

2

⎛
⎞
⎜ v
⎟ .
= ∑k =1 ⎜ Niki
⎜ ∑ vip ⎟⎟⎝ p =1 ⎠NiThen, the degree of cost-sharing δ i directly declines with the inequality of stakes ina group measured by the HHI. This observation suggests that, in the cases of constantelasticity of marginal costs, cost-sharing should be loosened up as the distribution of
stakes gets more unequal.
To examine this conjecture, we can use the concept of Lorenz-dominance todetermine whether a stake vector is more unequal (or “less nearly equal”) thananother vector. Take two stake vectors v = ( v1 ,!, vn ) and vʹ = ( v1ʹ ,!, vnʹ ). The latter
distribution of group members’ prize valuations is called more unequal than theformer in the sense of Lorenz-dominance, if ∑hk =1 vk ≥ ∑hk =1 vkʹ for all h ≤ n , with
31
This observation is true even in the case of observable cost-sharing rules. Hence we can generally
say that, as far as linear effort cost functions are assumed, the simple cost-sharing rules used in ourbasic model are sufficient to analyze the problem of intra-group heterogeneity of prize valuations.
More complicated rules are unnecessary.

27

strict inequality for at least one h, and ∑nk =1 vk = ∑nk =1 vkʹ . Straightforward calculations
are needed to confirm the following results regarding the relationship between the
Lehmer mean and this concept of inequality:32
Lemma 5. Let vʹi = ( viʹ1 ,!, vinʹ ) be a vector more unequal than v i = ( vi1 ,!, viN ) inithe sense of Lorenz-dominance. Then, for 2 ≥ q > 0 , Ln (vʹi , q ) > Ln (v i , q ).
Hence we can argue as follows; if αi ≥ 1 , or the marginal cost function is convex,
then 2 ≥ 1 + 1 and LN ⎛⎜ v i , 1 +1⎞⎟ is eligible for the above lemma: the more unequal
⎜ α
⎟
αi
i
⎝
⎠
i

stake vector in the sense of Lorenz dominance the group has, the larger the value ofthe Lehmer mean; that is, the degree of cost-sharing the group applies is reduced.
When the marginal cost function is strictly concave or 0 < αi < 1 , we do not obtain aclear-cut result as in the case where αi ≥ 1 . Since LN ⎛⎜ vi , 1 +1⎞⎟ is decreasing with
⎜ α
⎟
i
⎝
⎠
i

α i , however, the value of the discount factor γ i is never lower than the HHI if

0 < α i < 1. The HHI goes up and approaches 1 as the stake vector becomes moreunequal33. Thus the value of γ i asymptotically rises as the stake vector gets worse in
the sense of Lorenz dominance, even in the case of 0 < α i < 1. Thus our results
support the conjectures made at the beginning of this subsection. We summarize themas our last proposition.
Proposition 9. Assume that the effort cost function of an individual in group i has theform ci (a ) = K i ⋅ a

1+α i

1+ αi

( α i > 0 ). When αi ≥ 1 , a more unequal stake vector in thesense of Lorenz dominance results in a lower degree of cost-sharing. Even if
0 < αi < 1 , the degree of cost-sharing asymptotically declines as the distribution of
32

The statement “ v ʹ is more unequal than v in the sense of Lorenz-dominance” is equivalent to thestatement “v can be obtained from v' by a finite sequence of transformations (called the Daltontransfers)” of the form vil (t + 1) = vil (t ) + ε (t ) ≤ vih (t ), vih (t + 1) = vih (t ) − ε (t ) ≥ vil (t ) with ε (t ) > 0 forsome h and l ( h > l ), and vik (t +1) = vik (t ) for all k ≠ h, l , where at least in one of thetransformations the inequality must be strict. Lemma 4 is derived applying this property.
33
Since Lorenz-dominance is a partial ordering relation, we would need to adequately specify asequence of stake vectors.

28

stakes becomes more unequal.
6. Conclusion
Each group in a contest has to confront and manage its own collective-action problem,and devices of selective incentives are possible means of resolution. The widely
observed persistence of collective action problems suggests that those devices are not
perfect in resolving the problem. In this paper, we have argued that one of the reasons
is intra-group heterogeneity. When the disagreement on the valuation of the prize is
not negligible among the individual group members, the group faces the problem: towhat extent it should apply the device of selective incentives. Such a groupintrinsically has a problem of inefficient voluntary contributions, and selective
incentives can aggravate it.
We have examined how such a problem affects the adopted selective
incentive devices that take the form of cost-sharing rules. It has been found that therelative rate of change of marginal effort costs determines whether unequal valuationsof the prize in a group reduce the degree of cost-sharing. We have argued that usually
this rate would be strictly decreasing, and the equilibrium degree of cost-sharing in aheterogeneous group is lowered such that its winning probability is reduced
considerably. Even if the relative rate is increasing, groups with intra-groupheterogeneity at best choose the same cost-sharing rule as homogeneous groups. Our
results suggest therefore that the availability of selective incentives in a collective
contest is a cause of disadvantage to groups with intra-group heterogeneity.
Our conclusion is amended if a group can depend on the selective incentive
rule responding not only to the individual actions but also to their characteristics.
Actual rules of selective incentives may apply some degree of discrimination towarddifferent categories of individuals. How discriminating such rules are and how widely
they are adopted are interesting questions that deserve attention. Another topic forfurther study is the generalization of the group welfare function. If group leaders
embrace non-linear group welfare functions implying distributional concerns, howdoes it affect the equilibrium selective incentives? Are such concerns advantageous ordisadvantageous in enhancing the win probability of a competing group? Comparing
cost-sharing with prize-sharing under intra-group heterogeneity is also interesting.
Prize sharing is a much more popular type of device in the literature of group contests
to mitigate collective action problems. It seems natural to ask how the outcomes of29the two types of sharing rules differ. In particular, it is interesting to identify theconditions that justify the group’s selection of cost-sharing rather than prize-sharing.
We may also need to identify the conditions justifying the simultaneous (optimal) useof both sharing modes. A model of collective contests with prize-sharing rules
affording intra-group heterogeneity seems to be a prerequisite for such research.
7. Proofs
Proof of Lemma 1
The restrictions on the beliefs of the individuals by the “no-signaling-what-you-don’tknow” condition discussed in subsection 2. 2, the contribution by individual k ofgroup i at the information set indexed by δ i is described as the solution of

(

p≠k
*
j

aip* (δ i ) + a)⎧⎪
c aip* (δ i ) + ci (a )⎫⎪
∑
p≠k i
max
vik − ⎨ (1 − δ i )⋅ ci (a ) + δ i ⋅
⎬.
a ≥0
Ni
∑ j ≠i A*j (δ ) + ∑ p≠k aip* (δ i ) + a ⎪⎩⎪⎭∑Since aik* (δ i ) > 0 always holds due to the assumption: lim a→0 ciʹ(a) = 0 , equation (3) isthe first-order necessary and sufficient condition for aik* (δ i ) , the solution of thismaximization problem.
Q.E.D.
Proof of Lemma 2
The contribution of each individual in a group i , which is determined by equation (3),is increasing in δ i . The highest value of aggregate group effort is attained whenH
δ i = 1. Denoting this value by Ai , the contribution by individual k is derived fromthe equation

∑

j ≠i

j ≠i

j

Aj

v −
2 ik

(∑ A + A )Hi1ciʹ (aik ) = 0 , orNi⎡(ciʹ )−1 ⎢ Ni ⋅⎢⎣⎤⎥ = aik .
v
H 2 ik ⎥A+A∑ j ≠i j i
⎦

∑

(

j ≠i

Aj

)

The sum of the contributions by all members is equal to AiH . The definition of(

AiH ∑ j ≠i Aj) is thus derived. Similar arguments hold for A (∑ A ), by setting
L
i

j ≠i

j

δ i = 0 . Since the contribution by each individual is continuous in δ i , so is theaggregate group effort. Hence, the group leader can attain any value of group effort in

[ (

)

(

)]

the interval AiL ∑ Aj , AiH ∑ Aj , by the intermediate value theorem. The first
j ≠i
j ≠i
equation in (4) is straightforward. The second can be derived applying condition (3).
Q.E.D.

30

Proof of Lemma 3
By the conditions in (4), we have

∑

Ni

k =1

∂aik ( Ai ; v i )∂aik ( Ai ; v i ) vik ciʹʹ(ai1 ( Ai ; v i )) ∂ai1 ( Ai ; v i ).
= 1 and
=
⋅
⋅
∂Ai
∂Ai
vi1 ciʹʹ(aik ( Ai ; v i ))
∂Ai

Hence

vik
∂aik ( Ai ; v i )ciʹʹ(aik ( Ai ; v i ))for all k = 1,!, N i .
=
vip
Ni
∂Ai
∑ p=1 cʹʹ(a (A ; v ))
i
ip
i
i

(A1)

and we get equation (6) by definition of Ei ( Ai ; v i ).
Hence, 0 ≤ ∂ Ei ( Ai ; v i ) ≤ ∑ N ciʹ(aik ( Ai ; v i )) for all Ai > 0 . By Assumption 2, these
k =1
i

∂Ai

inequalities imply that lim A →0 ∂ Ei ( Ai ; v i ) = 0 .
i

Q.E.D.

∂Ai

Proof of Proposition 1
We will show that there exists a unique configuration of group efforts A*j , j = 1,!, msuch that Ai* is a solution of the maximization problem (7) for all i = 1,!, m .
Assuming the regularity condition, we can derive the following first-order necessary
and sufficient condition for the maximal group effort induced by the leader of group i:⎧ ∑ j ≠i A*j∂⎪
V −Ei Ai* ; v i ≤ 0 if Ai* < AiH ∑ j ≠i A*j
2 i
m
*
∂
A
⎪ ∑ Aji
j =1⎪⎨⎪ ∑ A*j
∂
j ≠i
⎪
V −
Ei Ai* ; v i ≥ 0 if AiL ∑ j ≠i A*j < Ai*
2 i
m
∂Ai
⎪
A*
⎩ ∑ j =1 j

(

)

(

)

(

(

(

)

(

)

)
(A2)

)

Let us apply the “share function” approach, originated by Esteban and Ray (2001) andCornes and Hartley (2005, 2007), to prove the existence and uniqueness ofequilibrium. To begin with, define the functions π iL ( A; v i ) and π iH ( A; v i ) by theequationsL⎧⎤Ni−1 ⎡ 1 − π i ( A; v i )ʹ()
c⋅ vik ⎥ = A ⋅ π iL ( A; v i )⎪ ∑k =1 i ⎢
A
⎣
⎦
⎪⎪
⎨
⎪ Ni
⎤
1 − π iH ( A; v i )−1 ⎡⋅ vik ⎥ = A ⋅ π iH ( A; v i )⎪ ∑k =1 (ciʹ ) ⎢ N i ⋅
A
⎪⎩
⎣
⎦

31

for A > 0 . Clearly, π iL ( A; v i ) < π iH ( A; v i ). Consider a configuration of group efforts
in which the sum of the effort put by all of the competing groups is A and also group i()

is constrained to solve problem (7) by the minimal feasible group effort, AiL ∑ Aj .
j ≠iThen the value of the group effort is given by A ⋅ π iL ( A; v i ). If the total effort is A andgroup i is constrained by the maximal feasible group effort, then group i’s effort isgiven by A ⋅ π iH ( A; v i ).
Also define the function π iUC ( A; v i ) by the equation(1 − π (A; v ))⋅V − A ⋅ ∂∂A E (A ⋅ π (A; v ); v ) = 0
UC
i

i

i

UC
i

i

i

(A3)

i

i

for A > 0 . This function is well defined because lim A →0 ∂ Ei ( Ai ; v i ) = 0 holds by∂Aii

Lemma 3. If the total effort is A and group i chooses an interior solution of (7), it isgiven by A ⋅ π iUC ( A; v i ) . Note that π iL ( A; v i ) , π iH ( A; v i ) and π iUC ( A; v i ) are allcontinuous and strictly decreasing in A > 0 . They also converge to 1 as A is reduced
to 0, and converge to 0 as A rises to infinity. Define the share function ofgroup i by

π i ( A; v i ) = max{ min {π iUC ( A; v i ), π iH ( A; v i ) }, π iL ( A; v i ) }for A > 0 . Then, π i ( A; v i ) is continuous and strictly decreasing in A > 0 . In addition,mlim A→0 π i ( A; v i ) = 1 and lim A→∞ π i ( A; v i ) = 0 . We can see that ∑ j =1 π j (A ; v j ) is

continuous, strictly decreasing in A > 0 , lim A→0 ∑mj=1π j ( A; v i ) = m > 1 andmlim A→∞ ∑ j =1π j ( A; v i ) = 0 . It should be clear that there exists a unique A* > 0 such thatm∑ π (A ; v ) = 1 for given v ,!, v
j =1

*

j

1

j

m

and ∑mj=1π j (A* ; v j )⋅ A* = A* . Denote

π j (A* ; v j )⋅ A* = A*j . Let us show that each Ai* is a solution of the maximization

problem (7) for all i = 1,!, m .
Consider the case where π i (A* ; v i ) = π iUC (A* ; v i ). Given the inequalities
π iL (A* ; v i ) ≤ π iUC (A* ; v i ) ≤ π iH (A* ; v i ),A* − Ai*
2( A − A + A (A − A ) )
*

*
i

L
i

*

*
i

if π iUC (A* ; v i )⋅ A* = Ai* < AiL (A* − Ai* ), then<A* − Ai*
* 2

(A )

≤

(

A* − A* ⋅ π iL A* ; v i
* 2

(A )

But then,

(

)

L
*
⎤
−1 ⎡ 1 − π i A ; v iʹA ≥ A ⋅ π A ; v i = ∑k =1 (ci ) ⎢⋅ vik ⎥
*
A
⎣
⎦
*
i

*

L
i

(

*

)

Ni

32

).

⎡
A* − Ai*N−1> ∑k =i 1 (ciʹ ) ⎢
*
*
L
*
*
⎣⎢ A − Ai + Ai A − Ai

(

(

⎤
L
*
*
⋅
v
⎥ = Ai A − Ai
ik
2
⎦⎥

(

))

)

and this contradiction proves that Ai* ≥ AiL (A* − Ai* ). In a similar way, we can provethat Ai* ≤ AiH (A* − Ai* ). Hence, Ai* = π iUC (A* ; v i )⋅ A* is an interior solution for the leaderof group i.
Consider the case where π i (A* ; v i ) = π iH (A* ; v i ). Given the inequalities
π iL (A* ; v i ) < π iH (A* ; v i ) ≤ π iUC (A* ; v i ), we can see that A* ⋅ π i (A* ; v i ) is a feasible choiceof the leader with full-sharing costs. Suppose that π iH (A* ; v i )⋅ A* = Ai* ≤ AiL (A* − Ai* ).
Then

A* − Ai*
2( A − A + A (A − A ) )
*

*
i

L
i

*

*
i

≤

A* − Ai*
* 2

(A )

<

(

A* − A* ⋅ π iL A* ; v i
* 2

)

(A )

and

(

)

L
*
⎤
N
−1 ⎡ 1 − π i A ; v i
Ai* > A* ⋅ π iL A* ; v i = ∑k =i 1 (ciʹ ) ⎢⋅ vik ⎥
*
A
⎣
⎦

(

)

⎡
A* − Ai*N−1> ∑k =i 1 (ciʹ ) ⎢⎢⎣ A* − Ai* + AiL A* − Ai*

(

(

⎤
L
*
*
⋅
v
ik ⎥ = Ai A − Ai .
2
⎥⎦

(

))

)

This contradiction proves that Ai* > AiL (A* − Ai* ). Since Ai* ≤ π iUC (A* ; v i )⋅ A* ,A* − Ai*∂∂Vi − A* ⋅Ei Ai* ; v i ≥ 1 − π iUC A* ; v i ⋅Vi − A* ⋅Ei A* ⋅ π iUC A* ; v i ; v i = 0
*
∂Ai
∂Ai
A

(

) (

(

))

(

(

) )

and Ai* satisfies the first-order condition (A2). Similarly, we can prove that, if
π i (A* ; v i ) = π iL (A* ; v i ), then it is a feasible choice with Ai* < AiH (A* − Ai* ) and, again, Ai*

satisfies the first-order condition (A1). In any case, therefore, Ai* is the best-responseof the group leader i for the maximization problem (7). The existence of equilibriumis therefore established.
Let us prove that the equilibrium is unique. It is clear that the total effort by
all of the competing groups must be positive in equilibrium. Let Aˆ1 ,!, Aˆ m be aconfiguration of the group efforts such that Âi is a solution of the maximization
problem (7) for all i = 1,!, m . Let Aˆ = ∑mj=1 Aˆ j > 0 .
If Âi is an unconstrained choice of effort by group i, it holds that

(

)

(

)

AiL Aˆ − Aˆ i ≤ Aˆ i ≤ AiH Aˆ − Aˆ i . We also have the equation33

Aˆ − Aˆ i∂
Vi −Ei Aˆi ; v i = 0 ,
2
∂Ai
ˆA

( )
()
which implies that Aˆ = Aˆ ⋅ π (Aˆ ; v ). Suppose that π (Aˆ ; v ) > π (Aˆ ; v ).
UC
i

i

UC
i

i

H
i

i

i

Then
Ni

−1

⎡

Aˆ − Aˆi⎤⎥ = AiH Aˆ − Aˆi ≥ Aˆ ⋅ π iUC Aˆ ; v i
⋅
v
ik
2
⎥⎦
A − Aˆi + AiH Ae − Aˆi∑ (cʹ ) ⎢⎢ N ⋅ ˆ
i

k =1

i

(

⎣

(

(

))

)

(

(

)

)

⎤
1 − π iH Aˆ ; v i
Ni
−1 ⎡
H ˆ
ˆ
ʹ
> A ⋅ π i A ; v i = ∑k =1 (ci ) ⎢ Ni ⋅⋅ vik ⎥ .
Aˆ
⎣
⎦

(

)

But our supposition implies that(Aˆ − Aˆi()

Aˆ − Aˆi 1 − π iH Aˆ ; v i.
≤<ˆˆA − Aˆ + AH Aˆ − Aˆ 2
ˆA 2
A
i
i
i

(

))

()

(

)

(

)

This contradiction proves that π iUC Aˆ ; v i ≤ π iH Aˆ ; v i . In a similar way, we can also(( ) ( ))

prove that π iUC Aˆ ; vi ≥ π iL Aˆ ; vi . Hence Aˆ i = Aˆ ⋅ π i Aˆ ; v i .

(

)

(

)

If Aˆi = AiL Aˆ − Aˆi , it must hold that Aˆi = Aˆ ⋅ π iL Aˆ ; v i . The first-order condition(A2) implies that()

1 − π iL Aˆ ; v i∂
Vi −Ei Aˆ ⋅ π iL Aˆ ; v i ; v i
ˆA
∂Ai

(

(

) )

(

)

UC ˆ
Aˆ − Aˆi∂ˆ ; v ≤ 0 = 1 − π i A; v i V − ∂ E Aˆ ⋅ π UC Aˆ ; v ; v .
V
−
E
A
i
i
i
i
i
i
i
i
i
2
∂Ai
∂Ai
Aˆ
Aˆ

=

(

()

(

)

(

)

)

(

(

) )

(

) ( )
Aˆ = A (Aˆ − Aˆ ) implies that π (Aˆ ; v ) ≤ π (Aˆ ; v ), and π (Aˆ ; v ) = π (Aˆ ; v ).
Now, we have proved that Aˆ = Aˆ ⋅ π (Aˆ ; v ) must hold for all j = 1,!, m . SinceAˆ = ∑ Aˆ > 0 , the equation ∑ π (Aˆ ; v ) = 1 holds. Â must therefore be uniqueand so is Aˆ = Aˆ ⋅ π (Aˆ ; v ) for all j = 1,!, m .
Hence, π iL Aˆ ; v i ≥ π iUC Aˆ ; v i and π iL Aˆ ; v i = π i Aˆ ; v i . Similarly we can prove that
i

H
i

H
i

i

UC
i

i

j

m

j =1

j

H
i

i

i

i

i

j

m

j

j =1

j

j

j

j

j

Q.E.D.
Proof of Proposition 3
Given the efforts by the other competing groups, the best configuration of thecontributions by the individuals in group i, aiB1 ,!, aiNB , maximizes the group objectivei34

functionNi∑ aV − ∑ c (a ) . Then ∑ c (a ) must be minimized to
A
+
a
∑
∑
k =1

i

Ni

j ≠i

j

Ni

Ni

ik

k =1

k =1 ik =1 i

ik

B
ik

ik

realize the best group effort AiB = ∑kN=1 aikB . It is necessary therefore that everyiindividual must bear the same marginal effort costs, i.e. ciʹ(aikB ) = ciʹ(aiB1 ) holds for allB

k = 1,!, Ni . The implication is that aikB = Ai

holds for all k = 1,!, Ni . Suppose thatNi

aikB can be induced by a sophisticated cost-sharing rule. The first-order condition∑ A ⋅ v − ⎛⎜1 − δ + δ ⎞⎟ ⋅ cʹ(a ) = 0 ,
j ≠i
2

j

ik

A

⎜
⎝

ik

ik

Ni ⎟⎠

i

B
ik

must hold for each individual k. The condition ciʹ(aikB ) = ciʹ(aiB1 ) is realized by setting

1 − δ ik +

δ ik

1 − δ i1 +

δ i1

Ni

=

Ni

vik
for all k = 1, !, Ni .
vi1

Since the best group effort AiB = ∑kN=1 aikB maximizesi(A4)⎛ AiB ⎞ , the
AiB
⎜
⎟⎟
V
−
N
⋅
c
i
i
i⎜
∑ j ≠i A j + AiB⎝ Ni ⎠

following first-order condition also holds:∑ A ⋅V − cʹ⎛⎜ A ⎞⎟ = 0 .
j ≠i
2

A

B
i

j

i

i

⎜N ⎟
⎝ i⎠

Comparing this equation with the first-order condition for individual k, we get thatv
1 − δ ik += ik
N i Vi

δ ikv
1 − ikor δ = Vi
ik
1
1−
Ni

for all k = 1,!, Ni . Since these equations arecompatible with equation (A4), we obtain the desired result.
Q.E.D.
Proof of Proposition 4
The first-order conditions (A2) for problem (7) is equivalent to the following
conditions:

35

vik
ciʹʹ(aik ( Ai ; v i ))NiNv − ∑k =i 1 ciʹ(aik ( Ai ; v i ))⋅≤ (≥ ) 0 ,2 ∑k =1 ik
m
vip
Ni
A
∑ j =1 j∑ p=1 cʹʹ(a (A ; v ))
i
ip
i
i

(

∑ j ≠i A j)

if δ i < 1 ( δ i > 0 ).
Since the contribution of each individual belonging to the group satisfies the equation∑ A ⋅ v = cʹ(a ( A ; v )) for all k = 1,!, N ,i(∑ A ) γ
j

j ≠i

ik

i

2

m

ik

i

i

i

j

j =1

we can derive the inequality in (11) by inserting these equations to the above
inequality. Let us turn to the equation in (11). Notice that the equilibrium
contributions by the individuals in group i satisfy the conditions in (4). In particular,( ) holds for all k = 1,!, N . Hence we obtain the equation:( )cʹ a *
vik = vi1 ⋅ i ik*
ciʹ ai1ivi1vikciʹ aik*ciʹ ai*1ciʹʹ aik*⋅=⋅∑k =1 Ni v
∑k =1 vi1 Ni *⎞∑ p =1 ip ∑ Np=i 1 ⎛⎜⎜ vip * ⎞⎟⎟cʹ aip ∑ N i ⎛⎜ vip* ∑ p =1 i
* ⎟
⎜
⎟
p
=
1
ʹ
c
a
i
i1
⎝ ciʹʹ aip ⎠⎝ ciʹʹ aip ⎠
Ni

vik

vik

( )

ciʹʹ aik*( ) ( )
( )
( )

Ni

( )

vik

Ni

( )
*
ik

∑k =1 ciʹ a ⋅
=

( )

( )

( )

ciʹʹ aik*⎞N ⎛v∑ p =i 1 ⎜⎜ ip ciʹʹ aip* ⎟⎟
⎝
⎠

Ni

( )

∑ p =1 ciʹ a

( ).

(A4)

*
ip

By using (6), the equation appearing in the condition (11) is directly obtained.
Q.E.D.
Proof of Proposition 5
Let xk =( )ciʹ aik*

vik

∑

Ni

v

and y =kp =1 ip( ) for k = 1,!, N .
⎛ cʹ (a )
⎞
⎟⎟
∑ ⎜⎜
ʹ
ʹ
(
)
c
a
⎝
⎠
Ni

p =1

i

ciʹʹ aik*
*
ip

i

i

*
ip

vik

Then,

Ni

Ni

k =1 kk =1∑ x =∑

yk = 1 and x1 ≤ ! ≤ x N i . Also y k =( )ciʹʹ aik*holds by⎛v⎞
∑ p =1 ⎜⎜ ip ciʹʹ aip* ⎟⎟
⎝
⎠
Ni

( )

the conditions in (4).
(a) Since ai*1 ≤ ! ≤ aiN* , we have y1 ≤ ! ≤ y N . Furthermore, intra-group heterogeneity
i

i

36

of the valuations of the prize implies that there exists some k such that xk < xk +1 andyk < yk +1 hold. Hence we can apply Chebyshev’s sum inequality to derive

∑

Ni
k =1

xk ⋅ y k >1 , Proposition 4 requires that1
1 . IfNNγi =⋅ ∑k =i 1 xk ⋅ ∑k =i 1 y k =
Ni
Ni
Ni

(

)(

)

Ni

γ i ≥ ∑k =1 xk ⋅ y k , contradicting the above strict inequality.

(b) In this case, y1 ≥ ! ≥ y N holds. Again by Chebyshev’s sum inequality,
i

∑

Ni
k =1

xk ⋅ y k ≤

1
1 .
N
N
⋅ ∑k =i 1 xk ⋅ ∑k =i 1 y k =
Ni
Ni

(

)(

)

If γ i > 1 , Proposition 4 requires γ i = ∑kN=1 xk ⋅ y , contradicting this inequality.
i

k

Ni

Q.E.D.
Proof of Proposition 6
Denote by 1 N the N i vector with all elements being equal to 1. When all individuals
iof group i have the same valuation of the prize, the stake vector can be represented byVi
1 N . For any degree of cost-sharing, they will choose the same effort level. For any
Ni ilevel of group effort Ai , therefore, the equation aik ⎛⎜ Ai ; Vi 1 N ⎞⎟ = Ai holds. We can
⎜
⎝

Ni

i

⎟
⎠

Ni

show that Ei ⎛⎜ Ai ; Vi 1 N ⎞⎟ = N i ⋅ ci ⎛⎜ Ai ⎞⎟ , and ∂Ei ⎛⎜ Ai ; Vi 1 N ⎞⎟ = ciʹ⎛⎜ Ai ⎞⎟ .
⎜
⎝

Ni

i

⎟
⎠

⎜N ⎟
⎝ i⎠

∂Ai ⎜⎝

Ni

i

⎟
⎠

⎜N ⎟
⎝ i⎠

Consider a stake vector vˆ i ≠ Vi 1 N . As the average of the prize valuations is constant,
Ni

i

Ni

∑ vˆ = V holds.
k =1 iki(a) When ciʹʹ(a ) is strictly decreasing, Corollary 2 and the convexity of ciʹ (a ) implyciʹ (a )⎛ˆ ⎞∂Ei(Ai ; vˆ i ) > 1 ∑kN=i 1 ciʹ(aik (Ai ; vˆ i )) ≥ ciʹ⎜⎜ ∑kN=i 1 aik (Ai ; vi ) ⎟⎟
∂Ai
Ni
Ni
⎝
⎠
⎛ A ⎞ ∂E ⎛ V⎞= ciʹ⎜⎜ i ⎟⎟ = i ⎜⎜ Ai ; i 1Ni ⎟⎟ .
⎝ N i ⎠ ∂Ai ⎝ N i⎠
Hence the marginal distorted group costs with the stake vector v̂ i is always larger
than that with Vi 1 N . On the other hand, the equilibrium degree of cost-sharing of
Ni

i

37

group i with the stake vector v̂ i is always strictly less than 1 by Proposition 5(a).
Hence, we can set π i ( A; vˆ i ) = π iUC ( A; vˆ i ) without loss of generality. The assumptionNi∑ vˆ = V and the equation (A3) imply thatk =1 iki⎛

π i ( A ; vˆ i ) = π iUC ( A; vˆ i ) < π iUC ⎜⎜ A ;
⎝

⎞
⎛ V
⎞
Vi
1 N i ⎟⎟ ≤ π i ⎜⎜ A ; i 1 N i ⎟⎟ for all A > 0 .
Ni
⎠
⎝ Ni
⎠

Let A* be the equilibrium total effort when the stake vector of group i isVi**
1 N i , and let A be the equilibrium effort when the stake vector of group i is v̂ i .
Ni

Remembering the necessary condition of the equilibrium total effort with the sharefunctions, we have

⎛

Vi

⎞

∑ π (A ; v )+ π (A ; vˆ ) = 1 = ∑ π (A ; v )+ π ⎜⎜ A ; N 1 ⎟⎟
**

j ≠i

**

j

j

i

*

i

j ≠i

j

j

i

*

Ni

⎝

i

(

) (

)

are

decreasingin⎠> ∑ j ≠i π j A* ; v j + π i A* ; vˆ i .
andbecause

A** < A*

the

share

functions

⎛

Vi

A.

Then,

⎞

∑ π (A ; v ) > ∑ π (A ; v ) and we get that π (A ; vˆ ) < π ⎜⎜ A ; N 1 ⎟⎟ .
**

j ≠i

j

*

j

j ≠i

j

**

j

i

i

i

⎝

*

Ni

i

⎠

(b) When ciʹʹ(a ) is increasing, a change in the stake vector from Vi 1 N to v̂ i inciʹ (a )Niigroup i does not lead to a change in the cost-sharing rule. The costs of individuals
belonging to the group are still fully shared. Then, the choices by individuals in thegroup obey the same first-order condition as in the case where they have the effort
cost function

1⋅ ci (a ) under no device of selective incentives. When ci (a ) isNistrictly convex, so is 1 ⋅ ci (a ). Hence we can apply a result by Nitzan and Ueda
Ni

(2013) on the relation of intra-group heterogeneity and strictly convex marginal effort
costs (their Proposition 4) to derive the desired result.
Q.E.D.
Proof of Proposition 7
Since the contributions induced under a cost-sharing rule are the same as in theoriginal model, we have (A1) as the relation between each individual’s contribution
and the aggregate effort intended by the group leader. Because of the convexity of38

Eˆ i ( Ai ; v i ),0=

∑ j ≠i A j
A2

N i − Eˆ iʹ( Ai ; v i ) =∑ j ≠i A j
A2vikciʹʹ(aik ( Ai ; v i ))N cʹ (a ( A ; v ))N i − ∑k =i 1 i ik i i ⋅v
Nvik∑ p =i 1 cʹʹ(a (ipA ; v ))
i
ip
i
i

is a first-order necessary and sufficient condition for the first-best group effort.
The contribution of each individual belonging to the group satisfies the equation

∑

j ≠i
2

Aj

= γi ⋅

A1⋅ ciʹ(aik ( Ai ; v i )) for all k = 1,!, Ni .
vik

Hence, the leader induces the first-best group effort by setting γ i =1.
Ni
Q.E.D.

Proof of Proposition 8
The Lehmer mean has the following properties. (i) Ln (v, q ) is strictly increasing in qnv

given v; (ii) Ln (v, 1) = ∑ k =1 k ; and (iii) lim q →∞ Ln (v, q ) = max{v1 , …, vn }. Applying thesenproperties to the result of Lemma 4, i.e. equation (13′), the proof of the claims in
Proposition 8 are directly derived.

Q.E.D.

39

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